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<li><a class="reference internal" href="#">Compressive sensing: tomography reconstruction with L1 prior (Lasso)</a></li>
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  <div class="sphx-glr-download-link-note admonition note">
<p class="admonition-title">Note</p>
<p>Click <a class="reference internal" href="#sphx-glr-download-auto-examples-applications-plot-tomography-l1-reconstruction-py"><span class="std std-ref">here</span></a> to download the full example code or to run this example in your browser via Binder</p>
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<div class="sphx-glr-example-title section" id="compressive-sensing-tomography-reconstruction-with-l1-prior-lasso">
<span id="sphx-glr-auto-examples-applications-plot-tomography-l1-reconstruction-py"></span><h1>Compressive sensing: tomography reconstruction with L1 prior (Lasso)<a class="headerlink" href="#compressive-sensing-tomography-reconstruction-with-l1-prior-lasso" title="Permalink to this headline">¶</a></h1>
<p>This example shows the reconstruction of an image from a set of parallel
projections, acquired along different angles. Such a dataset is acquired in
<strong>computed tomography</strong> (CT).</p>
<p>Without any prior information on the sample, the number of projections
required to reconstruct the image is of the order of the linear size
<code class="docutils literal notranslate"><span class="pre">l</span></code> of the image (in pixels). For simplicity we consider here a sparse
image, where only pixels on the boundary of objects have a non-zero
value. Such data could correspond for example to a cellular material.
Note however that most images are sparse in a different basis, such as
the Haar wavelets. Only <code class="docutils literal notranslate"><span class="pre">l/7</span></code> projections are acquired, therefore it is
necessary to use prior information available on the sample (its
sparsity): this is an example of <strong>compressive sensing</strong>.</p>
<p>The tomography projection operation is a linear transformation. In
addition to the data-fidelity term corresponding to a linear regression,
we penalize the L1 norm of the image to account for its sparsity. The
resulting optimization problem is called the <a class="reference internal" href="../../modules/linear_model.html#lasso"><span class="std std-ref">Lasso</span></a>. We use the
class <a class="reference internal" href="../../modules/generated/sklearn.linear_model.Lasso.html#sklearn.linear_model.Lasso" title="sklearn.linear_model.Lasso"><code class="xref py py-class docutils literal notranslate"><span class="pre">sklearn.linear_model.Lasso</span></code></a>, that uses the coordinate descent
algorithm. Importantly, this implementation is more computationally efficient
on a sparse matrix, than the projection operator used here.</p>
<p>The reconstruction with L1 penalization gives a result with zero error
(all pixels are successfully labeled with 0 or 1), even if noise was
added to the projections. In comparison, an L2 penalization
(<a class="reference internal" href="../../modules/generated/sklearn.linear_model.Ridge.html#sklearn.linear_model.Ridge" title="sklearn.linear_model.Ridge"><code class="xref py py-class docutils literal notranslate"><span class="pre">sklearn.linear_model.Ridge</span></code></a>) produces a large number of labeling
errors for the pixels. Important artifacts are observed on the
reconstructed image, contrary to the L1 penalization. Note in particular
the circular artifact separating the pixels in the corners, that have
contributed to fewer projections than the central disk.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="vm">__doc__</span><span class="p">)</span>

<span class="c1"># Author: Emmanuelle Gouillart &lt;emmanuelle.gouillart@nsup.org&gt;</span>
<span class="c1"># License: BSD 3 clause</span>

<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">from</span> <span class="nn">scipy</span> <span class="kn">import</span> <span class="n">sparse</span>
<span class="kn">from</span> <span class="nn">scipy</span> <span class="kn">import</span> <span class="n">ndimage</span>
<span class="kn">from</span> <span class="nn">sklearn.linear_model</span> <span class="kn">import</span> <span class="n">Lasso</span>
<span class="kn">from</span> <span class="nn">sklearn.linear_model</span> <span class="kn">import</span> <span class="n">Ridge</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>


<span class="k">def</span> <span class="nf">_weights</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">dx</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">orig</span><span class="o">=</span><span class="mi">0</span><span class="p">):</span>
    <span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">ravel</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
    <span class="n">floor_x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">floor</span><span class="p">((</span><span class="n">x</span> <span class="o">-</span> <span class="n">orig</span><span class="p">)</span> <span class="o">/</span> <span class="n">dx</span><span class="p">)</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">int64</span><span class="p">)</span>
    <span class="n">alpha</span> <span class="o">=</span> <span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">orig</span> <span class="o">-</span> <span class="n">floor_x</span> <span class="o">*</span> <span class="n">dx</span><span class="p">)</span> <span class="o">/</span> <span class="n">dx</span>
    <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">hstack</span><span class="p">((</span><span class="n">floor_x</span><span class="p">,</span> <span class="n">floor_x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)),</span> <span class="n">np</span><span class="o">.</span><span class="n">hstack</span><span class="p">((</span><span class="mi">1</span> <span class="o">-</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">alpha</span><span class="p">))</span>


<span class="k">def</span> <span class="nf">_generate_center_coordinates</span><span class="p">(</span><span class="n">l_x</span><span class="p">):</span>
    <span class="n">X</span><span class="p">,</span> <span class="n">Y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mgrid</span><span class="p">[:</span><span class="n">l_x</span><span class="p">,</span> <span class="p">:</span><span class="n">l_x</span><span class="p">]</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">float64</span><span class="p">)</span>
    <span class="n">center</span> <span class="o">=</span> <span class="n">l_x</span> <span class="o">/</span> <span class="mf">2.</span>
    <span class="n">X</span> <span class="o">+=</span> <span class="mf">0.5</span> <span class="o">-</span> <span class="n">center</span>
    <span class="n">Y</span> <span class="o">+=</span> <span class="mf">0.5</span> <span class="o">-</span> <span class="n">center</span>
    <span class="k">return</span> <span class="n">X</span><span class="p">,</span> <span class="n">Y</span>


<span class="k">def</span> <span class="nf">build_projection_operator</span><span class="p">(</span><span class="n">l_x</span><span class="p">,</span> <span class="n">n_dir</span><span class="p">):</span>
    <span class="sd">&quot;&quot;&quot; Compute the tomography design matrix.</span>

<span class="sd">    Parameters</span>
<span class="sd">    ----------</span>

<span class="sd">    l_x : int</span>
<span class="sd">        linear size of image array</span>

<span class="sd">    n_dir : int</span>
<span class="sd">        number of angles at which projections are acquired.</span>

<span class="sd">    Returns</span>
<span class="sd">    -------</span>
<span class="sd">    p : sparse matrix of shape (n_dir l_x, l_x**2)</span>
<span class="sd">    &quot;&quot;&quot;</span>
    <span class="n">X</span><span class="p">,</span> <span class="n">Y</span> <span class="o">=</span> <span class="n">_generate_center_coordinates</span><span class="p">(</span><span class="n">l_x</span><span class="p">)</span>
    <span class="n">angles</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">,</span> <span class="n">n_dir</span><span class="p">,</span> <span class="n">endpoint</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
    <span class="n">data_inds</span><span class="p">,</span> <span class="n">weights</span><span class="p">,</span> <span class="n">camera_inds</span> <span class="o">=</span> <span class="p">[],</span> <span class="p">[],</span> <span class="p">[]</span>
    <span class="n">data_unravel_indices</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">l_x</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>
    <span class="n">data_unravel_indices</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">hstack</span><span class="p">((</span><span class="n">data_unravel_indices</span><span class="p">,</span>
                                      <span class="n">data_unravel_indices</span><span class="p">))</span>
    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">angle</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">angles</span><span class="p">):</span>
        <span class="n">Xrot</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">angle</span><span class="p">)</span> <span class="o">*</span> <span class="n">X</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">angle</span><span class="p">)</span> <span class="o">*</span> <span class="n">Y</span>
        <span class="n">inds</span><span class="p">,</span> <span class="n">w</span> <span class="o">=</span> <span class="n">_weights</span><span class="p">(</span><span class="n">Xrot</span><span class="p">,</span> <span class="n">dx</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">orig</span><span class="o">=</span><span class="n">X</span><span class="o">.</span><span class="n">min</span><span class="p">())</span>
        <span class="n">mask</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">logical_and</span><span class="p">(</span><span class="n">inds</span> <span class="o">&gt;=</span> <span class="mi">0</span><span class="p">,</span> <span class="n">inds</span> <span class="o">&lt;</span> <span class="n">l_x</span><span class="p">)</span>
        <span class="n">weights</span> <span class="o">+=</span> <span class="nb">list</span><span class="p">(</span><span class="n">w</span><span class="p">[</span><span class="n">mask</span><span class="p">])</span>
        <span class="n">camera_inds</span> <span class="o">+=</span> <span class="nb">list</span><span class="p">(</span><span class="n">inds</span><span class="p">[</span><span class="n">mask</span><span class="p">]</span> <span class="o">+</span> <span class="n">i</span> <span class="o">*</span> <span class="n">l_x</span><span class="p">)</span>
        <span class="n">data_inds</span> <span class="o">+=</span> <span class="nb">list</span><span class="p">(</span><span class="n">data_unravel_indices</span><span class="p">[</span><span class="n">mask</span><span class="p">])</span>
    <span class="n">proj_operator</span> <span class="o">=</span> <span class="n">sparse</span><span class="o">.</span><span class="n">coo_matrix</span><span class="p">((</span><span class="n">weights</span><span class="p">,</span> <span class="p">(</span><span class="n">camera_inds</span><span class="p">,</span> <span class="n">data_inds</span><span class="p">)))</span>
    <span class="k">return</span> <span class="n">proj_operator</span>


<span class="k">def</span> <span class="nf">generate_synthetic_data</span><span class="p">():</span>
    <span class="sd">&quot;&quot;&quot; Synthetic binary data &quot;&quot;&quot;</span>
    <span class="n">rs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">RandomState</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
    <span class="n">n_pts</span> <span class="o">=</span> <span class="mi">36</span>
    <span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">ogrid</span><span class="p">[</span><span class="mi">0</span><span class="p">:</span><span class="n">l</span><span class="p">,</span> <span class="mi">0</span><span class="p">:</span><span class="n">l</span><span class="p">]</span>
    <span class="n">mask_outer</span> <span class="o">=</span> <span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">l</span> <span class="o">/</span> <span class="mf">2.</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span> <span class="o">+</span> <span class="p">(</span><span class="n">y</span> <span class="o">-</span> <span class="n">l</span> <span class="o">/</span> <span class="mf">2.</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span> <span class="o">&lt;</span> <span class="p">(</span><span class="n">l</span> <span class="o">/</span> <span class="mf">2.</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span>
    <span class="n">mask</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">l</span><span class="p">,</span> <span class="n">l</span><span class="p">))</span>
    <span class="n">points</span> <span class="o">=</span> <span class="n">l</span> <span class="o">*</span> <span class="n">rs</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">n_pts</span><span class="p">)</span>
    <span class="n">mask</span><span class="p">[(</span><span class="n">points</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">int</span><span class="p">),</span> <span class="p">(</span><span class="n">points</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">int</span><span class="p">)]</span> <span class="o">=</span> <span class="mi">1</span>
    <span class="n">mask</span> <span class="o">=</span> <span class="n">ndimage</span><span class="o">.</span><span class="n">gaussian_filter</span><span class="p">(</span><span class="n">mask</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="n">l</span> <span class="o">/</span> <span class="n">n_pts</span><span class="p">)</span>
    <span class="n">res</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">logical_and</span><span class="p">(</span><span class="n">mask</span> <span class="o">&gt;</span> <span class="n">mask</span><span class="o">.</span><span class="n">mean</span><span class="p">(),</span> <span class="n">mask_outer</span><span class="p">)</span>
    <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">logical_xor</span><span class="p">(</span><span class="n">res</span><span class="p">,</span> <span class="n">ndimage</span><span class="o">.</span><span class="n">binary_erosion</span><span class="p">(</span><span class="n">res</span><span class="p">))</span>


<span class="c1"># Generate synthetic images, and projections</span>
<span class="n">l</span> <span class="o">=</span> <span class="mi">128</span>
<span class="n">proj_operator</span> <span class="o">=</span> <span class="n">build_projection_operator</span><span class="p">(</span><span class="n">l</span><span class="p">,</span> <span class="n">l</span> <span class="o">//</span> <span class="mi">7</span><span class="p">)</span>
<span class="n">data</span> <span class="o">=</span> <span class="n">generate_synthetic_data</span><span class="p">()</span>
<span class="n">proj</span> <span class="o">=</span> <span class="n">proj_operator</span> <span class="o">*</span> <span class="n">data</span><span class="o">.</span><span class="n">ravel</span><span class="p">()[:,</span> <span class="n">np</span><span class="o">.</span><span class="n">newaxis</span><span class="p">]</span>
<span class="n">proj</span> <span class="o">+=</span> <span class="mf">0.15</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="o">*</span><span class="n">proj</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>

<span class="c1"># Reconstruction with L2 (Ridge) penalization</span>
<span class="n">rgr_ridge</span> <span class="o">=</span> <span class="n">Ridge</span><span class="p">(</span><span class="n">alpha</span><span class="o">=</span><span class="mf">0.2</span><span class="p">)</span>
<span class="n">rgr_ridge</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">proj_operator</span><span class="p">,</span> <span class="n">proj</span><span class="o">.</span><span class="n">ravel</span><span class="p">())</span>
<span class="n">rec_l2</span> <span class="o">=</span> <span class="n">rgr_ridge</span><span class="o">.</span><span class="n">coef_</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">l</span><span class="p">,</span> <span class="n">l</span><span class="p">)</span>

<span class="c1"># Reconstruction with L1 (Lasso) penalization</span>
<span class="c1"># the best value of alpha was determined using cross validation</span>
<span class="c1"># with LassoCV</span>
<span class="n">rgr_lasso</span> <span class="o">=</span> <span class="n">Lasso</span><span class="p">(</span><span class="n">alpha</span><span class="o">=</span><span class="mf">0.001</span><span class="p">)</span>
<span class="n">rgr_lasso</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">proj_operator</span><span class="p">,</span> <span class="n">proj</span><span class="o">.</span><span class="n">ravel</span><span class="p">())</span>
<span class="n">rec_l1</span> <span class="o">=</span> <span class="n">rgr_lasso</span><span class="o">.</span><span class="n">coef_</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">l</span><span class="p">,</span> <span class="n">l</span><span class="p">)</span>

<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="mf">3.3</span><span class="p">))</span>
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">131</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="n">plt</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">gray</span><span class="p">,</span> <span class="n">interpolation</span><span class="o">=</span><span class="s1">&#39;nearest&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="s1">&#39;off&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">&#39;original image&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">132</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">rec_l2</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="n">plt</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">gray</span><span class="p">,</span> <span class="n">interpolation</span><span class="o">=</span><span class="s1">&#39;nearest&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">&#39;L2 penalization&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="s1">&#39;off&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">133</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">rec_l1</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="n">plt</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">gray</span><span class="p">,</span> <span class="n">interpolation</span><span class="o">=</span><span class="s1">&#39;nearest&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">&#39;L1 penalization&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="s1">&#39;off&#39;</span><span class="p">)</span>

<span class="n">plt</span><span class="o">.</span><span class="n">subplots_adjust</span><span class="p">(</span><span class="n">hspace</span><span class="o">=</span><span class="mf">0.01</span><span class="p">,</span> <span class="n">wspace</span><span class="o">=</span><span class="mf">0.01</span><span class="p">,</span> <span class="n">top</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">bottom</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">left</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span>
                    <span class="n">right</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>

<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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